Near orderings of topological spaces (Q793390)

Starting with a topological space X and an equivalence relation \(\sim\) on X, a ''compatible near order'' is defined to be a transitive relation \(<\) which satisfies certain compatibility conditions with both the topology and the equivalence relation. This paper is concerned with the existence of such a compatible near order, and, in case it exists, with the existence of an order-preserving, real-valued function on the space which is constant on each equivalence class. Both problems turn out to have simple solutions subject to certain mild conditions on X and \(\sim\).